$\overline{AB} = \sqrt{74}$ $\overline{BC} = {?}$ $A$ $C$ $B$ $\sqrt{74}$ $?$ $ \sin( \angle ABC ) = \frac{5\sqrt{74} }{74}, \cos( \angle ABC ) = \frac{7\sqrt{74} }{74}, \tan( \angle ABC ) = \dfrac{5}{7}$
$\overline{AB}$ is the hypotenuse $\overline{BC}$ is adjacent to $\angle ABC$ SOH CAH TOA We know the hypotenuse and need to solve for the adjacent side so we can use the cos function (CAH) $ \cos( \angle ABC ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\overline{BC}}{\overline{AB}}= \frac{\overline{BC}}{\sqrt{74}} $ $ \overline{BC}=\sqrt{74} \cdot \cos( \angle ABC ) = \sqrt{74} \cdot \frac{7\sqrt{74} }{74} = 7$